Multi-adjoint property-oriented and object-oriented concept lattices

Information Sciences 190 (2012) 95–106

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Information Sciences

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Multi-adjoint property-oriented and object-oriented concept lattices

Jesús Medina ⇑Department of Mathematics, University of Cádiz, Spain

a r t i c l e i n f o

Article history:Received 3 December 2010Received in revised form 19 September 2011Accepted 11 November 2011Available online 25 November 2011

Keywords:Formal concept latticeRough setsFuzzy sets

0020-0255/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.ins.2011.11.016

⇑ Fax: +34 952 132766.E-mail address: [email protected]

a b s t r a c t

This paper presents a generalisation of the classical property and object-oriented conceptlattices to a fuzzy environment based on the philosophy of the multi-adjoint paradigm.These concept lattices are generalisations of rough set theory used to consider two differ-ent sets – the set of objects and the set of attributes – to apply the corresponding modaloperators, as in formal concept analysis.

First of all, the paper presents several specific properties of adjoint triples in which theduals of several supports are considered. These properties are then used to introduce themulti-adjoint property and object-oriented concept lattices, in which different adjointtriples can be used in non-linear sets, as well as the corresponding representation (funda-mental) theorems. Moreover, these lattices are related to the multi-adjoint concept lattice,in which negation operators are not needed; as a result, this relation allows the propertiesto be translated from one to another.

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1. Introduction

Formal concept analysis and rough set theory are important tools in furthering knowledge of relational informationsystems.

Rough set theory was originally proposed by Pawlak [25] as a formal tool for modelling and processing incompleteinformation in information systems. This theory was extended by Düntsch and Gediga in [11,13] in order to consider twodifferent sets: the set of objects and the set of attributes. This extension is called property-oriented concept lattice [6].

Fuzzy sets and rough sets have been related and a number of fuzzy extensions of rough set theory and property-orientedconcept lattices have been presented in order to represent and analyse both incomplete information and imprecise informa-tion. In [27], the authors introduced ðI ; T Þ fuzzy rough sets in which I is an S-, R- or QL-implication, which satisfies severalproperties, and T is a t-norm, which extends the fuzzy rough set framework given in [10]. In the residuated case, this was inturn embedded by the fuzzy framework of the property-oriented concept lattice presented in [14,18].

Furthermore, formal concept analysis, introduced by Wille in [29], is another useful tool for qualitative data analysis andhas become an appealing major research topic, from both the theoretical and applied perspectives.

The literature presents different fuzzy extensions of formal concept analysis. To the best of our knowledge, the first exten-sion was given in [4], although it did not advance far beyond the basic definitions, probably because residuated implicationswere not used. Subsequently, in [2,26], the authors independently used complete residuated lattices as structures for thetruth degrees; for this approach, a representation theorem was proved directly in a fuzzy framework, which establishedthe basis of most of the subsequent direct proofs.

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96 J. Medina / Information Sciences 190 (2012) 95–106

Multi-adjoint concept lattices were introduced [22] as a new general approach to formal concept analysis, in which thephilosophy of the multi-adjoint paradigm was applied to the formal concept analysis (see [16,24] for further information), inorder to provide a general framework that could conveniently accommodate the different approaches mentioned above. Theauthors worked in a general non-commutative environment and this naturally led to the consideration of several adjoint tri-ples, also known as implication triples [1], as the main building blocks of a multi-adjoint concept lattice. As a result, differentdegrees of preference related to the set of objects or attributes can be easily established.

Several papers [19,21,32,33] have related these two important tools – formal concept analysis and rough set theory – inthe classical case. Consequently, the results presented in the formal concept analysis framework can be applied to rough settheory and vice versa [6,28]. In the fuzzy case, Georgescu and Popescu [14], and, more recently, Djouadi and Prade [8], haverelated the derivation operators in a fuzzy level although, for example, they use negation operators.

The paper presents a number of properties related to adjoint triples that were obtained when dual lattices are considered.These results will be used to generalise the classical property-oriented and object-oriented concept lattices [6] to a fuzzyenvironment based on the philosophy of the multi-adjoint paradigm [23].

Moreover, these definitions generalise the fuzzy definitions given in [14,18,27], in the residuated case, and the theorydeveloped from one kind of concept lattice can be applied to the others. For example, in this paper, the fundamental theoremof the multi-adjoint concept lattice [22] has been changed to obtain a fundamental theorem for the multi-adjoint property-oriented and object-oriented concept lattice.

This paper is structured as follows: Section 2 reviews the different ‘‘crisp’’ definitions of derivation operators that formdifferent concept lattices and several other results, which will be used later; a number of adjoint triple properties are intro-duced in Section 3 and a summary of the multi-adjoint concept lattices is provided in [22], which is used in Sections 4 and 5to present the multi-adjoint property and object-oriented concept lattices together with the corresponding representationtheorems; Section 6 is a detailed example of multi-adjoint property concept lattices and includes a comparison with themulti-adjoint concept lattices. Finally, the paper ends with several conclusions and prospects for future work.

2. Formal concept analysis: classical derivation operators

The intension and extension mappings in formal concept analysis form a derivation operator, which is a Galois connec-tion, although there exists other mappings that form other derivation operators. In this section, a short introduction of for-mal concept analysis and other derivation operators will be presented.

In classical formal concept analysis, we consider a set of attributes A, a set of objects B and a crisp relation between themR :A � B ? {0,1}, where, for each a 2 A and b 2 B, we have that R(a,b) = 1, if a and b are related, otherwise R(a,b) = 0. We willalso write aRb when R(a,b) = 1. The triple (A,B,R) is called context and the mappings M :2B ? 2A, M :2A ? 2B, are defined, foreach X # B and Y # A, as:

XM ¼ fa 2 Aj for all b 2 X; aRbg ¼ fa 2 Aj if b 2 X; then aRbg ð1ÞYM ¼ fb 2 Bj for all a 2 Y; aRbg ¼ fb 2 Bj if a 2 Y; then aRbg ð2Þ

A concept in the context (A,B,R) is defined to be a pair (X,Y), where X # B, Y # A, and which satisfies that XM = Y andYM = X. The element X of the concept (X,Y) is the extent and Y the intent.

The set of concepts in a context (A,B,R) is denoted as BðA;B;RÞ and is a complete lattice [12,7], with the inclusionordering on the left argument or the opposite of the inclusion ordering on the right argument, that is, givenðX1;Y1Þ; ðX2;Y2Þ 2 BðA;B;RÞ, we have that (X1,Y1) 6 (X2,Y2) if X1 # X2 (or, equivalently, Y2 # Y1).

An important fact is that the mappings M :2B ? 2A and M :2A ? 2B form a Galois connection [7]. There are two dual versionsof this definition. The one we adopt here is the most famous Galois connection of all, that discovered by Galois, where themaps are order-reversing, which will be properly called the Galois connection, and the other in which the maps are order-preserving, that will be called isotone Galois connection. There are arguments for both versions, although, at a theoretical le-vel, the difference is not significant since we can pass from one to another swapping a lattice by its dual, for example, 2B by(2B)@.

In order to make this contribution self-contained, we recall its formal definitions below:

Definition 1. Let (P1,61) and (P2,62) be posets, and ; :P1 ? P2," :P2 ? P1 mappings, the pair (", ;) forms a Galois connectionbetween P1 and P2 if and only if:

1. " and ; are order-reversing.2. x 61 x;" for all x 2 P1.3. y 62 y"; for all y 2 P2.

Definition 2. Let (P1,61) and (P2,62) be posets, and ; :P1 ? P2," :P2 ? P1 mappings, the pair (", ;) forms an isotone Galois con-nection between P1 and P2 if and only if:

J. Medina / Information Sciences 190 (2012) 95–106 97

1. " and ; are order-preserving.2. x 61 x;" for all x 2 P1.3. y"; 62 y for all y 2 P2.

This last definition arises from the notion of residuated mappings, that is, the mappings of an isotone Galois connectionare residuated mappings of each other, since the definition of residuated mappings is equivalent to the one given to isotoneGalois connection, indeed, some times in the literature, isotone Galois connection are called residuated mappings.

Before following with the comments, we need to recall the definition of opposite order. Given a set P and an order rela-tion, 6, on P, the opposite order (also dual, inverse, or converse, etc.) of 6 is the relation 6op, defined as x1 6

opx2 if and only ifx2 6 x1, for all x1, x2 2 P. Usually, we will write P instead of the partially ordered set (P,6), P@ instead of (P,6op), and we willsay that P@ is the dual of P.

Now, as we remarked above, the definition of isotone Galois connection follows from the original one considering P@2 in-stead of P2. Hence, an isotone Galois connection (", ;) on P1 and P2 is a Galois connection on P1 and P@2, and we can translate theproperties of Galois connections to isotone Galois connections.

A direct consequence of the definition of Galois connection is that x; = x;";, y" = y";", for all x 2 P1 and y 2 P2. Hence, allconcepts in BðA;B;RÞ are pairs (XMM,XM), where X # B, or (YM,YMM), where Y # A, that is:

BðA;B;RÞ ¼ fðXMM;XMÞj X # Bg ¼ fðYM;YMMÞj Y # Ag

Another interesting property is that, given a Galois connection (", ;), the mappings ;", "; are closure operators:

Definition 3. Let (P,6) be a poset, c :P ? P is a closure operator if it is increasing, x 6 c(x) and c(c(x)) = c(x), for all x 2 P.

Hence, the mappings MM :2A ? 2A and MM :2B ? 2B are closure operators too, and, hence, their fixed points are obtained di-rectly, which are the extents and the intents of the concepts in BðA;B;RÞ, respectively.

In order to obtain fix points, there exists another property which allows us to calculate fix points as easily as a closureoperator, that is, if the operator is an interior operator.

Definition 4. Let (P,6) be a poset, int:P ? P is a interior operator if it is decreasing, int(x) 6 x and int(int(x)) = int(x), for allx 2 P.

The above definition of extent and intent operators is a classical definition but it is not the only one. There exists threemore definitions considered in several frameworks: qualitative data analysis [13,11], crisp rough set theory [33], fuzzy roughset theory [5,20]. Some extra motivations about these operators are also introduced in [9,31,27,18,14].

Given the sets A, B, and a crisp relation R :A � B ? {0,1}, we have the mappings p :2B ? 2A, N :2B ? 2A, r :2B ? 2A defined,for each X # B, as:

Xp ¼ fa 2 Aj there exists b 2 X; such that aRbg ð3ÞXN ¼ fa 2 Aj for all b 2 B; if aRb; then b 2 Xg ð4ÞXr ¼ fa 2 Aj there exists b 2 Xc; such that aRcbg ð5Þ

where Xc and Rc mean the complement of X and the complement relation of R.Analogously, abusing of notation, we can define the mappings: p :2A ? 2B, N :2A ? 2B and r :2A ? 2B.These operators are called possibility, necessity and dual sufficiency operators, respectively; the classical one is called suf-

ficient operator. They are considered by pairs in order to form Galois connections or closure operators [12,29,32,6,13,11]. As aconsequence, several concept lattices are obtained: the classical formal concept lattice, dual formal concept lattice, property-ori-ented concept lattice and object-oriented concept lattice [6].

Clearly, the dual sufficiency operator satisfies that: Xr = ((Xc)M)c, for each X # B, therefore these operators are not inde-pendent and the concept lattice given from them are related, specifically we can obtain one from the other.

Moreover, the necessity and possibility operators are related to the sufficient operators, for details see [6,18].

3. Adjoint triples and multi-adjoint concept lattices

Assuming non-commutativity on the conjunctor directly provides two different ways of generalising the well-known ad-joint property between a t-norm and its residuated implication [15], depending on which argument is fixed.

Definition 5. Let (P1,61), (P2,62), (P3,63) be posets and & :P1 � P2 ? P3,. :P3 � P2 ? P1,- :P3 � P1 ? P2 be mappings, then(&,.,-) is an adjoint triple with respect to P1, P2, P3 if:

1. & is order-preserving in both arguments.2. . and - are order-preserving on the first argument and order-reversing on the second argument.3. x 61 z.y iff x & y 63 z iff y 62 z-x, where x 2 P1, y 2 P2 and z 2 P3.


Before defining the multi-adjoint concept lattices, we will introduce an interesting result about adjoint triples that wewill use later to define the multi-adjoint property-oriented and object-oriented concept lattices. Specifically, this result stud-ies what happens if we change the original order on the posets by the opposite one.

Lemma 1. Given the posets (P1,61), (P2,62), (P3,63) and an adjoint triple (&,.,-) with respect them, we obtain that:

1. (&op,-,.) is an adjoint triple with respect to P2, P1, P3.2. (.,&,-op) is an adjoint triple with respect to P@3; P2; P

@1.

3. (-,&op,.op) is an adjoint triple with respect to P@3; P1; [email protected]. (-op,.op,&op) is an adjoint triple with respect to P1; P@3; P@2.

Proof

(1) Follows directly from Definition 5.(2) Since (&,.,-) is an adjoint triple with respect to (P1,61), (P2,62), (P3,63), we have that: &:

(P1,61) � (P2,62) ? (P3,63), is increasing on both arguments, and . : (P3,63) � (P2,62) ? (P1,61), -:(P3,63) � (P1,61) ? (P2,62), are increasing on the first argument and decreasing on the second. Hence, we obtain that.: P3;6

op3

� �� ðP2;62Þ ! P1;6

op1

� �, is increasing on both arguments, and & : P1;6

op1

� �� ðP2;62Þ !

P3;6op3

� �;-op : P1;6

op1

� �� P3;6

op3

� �! ðP2;62Þ, are increasing on the first argument and decreasing on the second.

Moreover, for all x 2 P1, y 2 P2 and z 2 P3, we have that:

1 A siframe.

x61z. y iff x&y63z iff y62z- x

which is equivalent to:

z. y6op1 x iff z6op

3 x&y iff y62x-opz

that is:

z6op3 x&y iff z. y6op

1 x iff y62x-opz

which leads us to check that (.,&,-op) is an adjoint triple with respect to P3;6op3

� �; ðP2;62Þ; P1;6

op1

� �, that is, P@3; P2 and P@1.

(3) follows similarly to item (2).(4) is a direct consequence of items (1) and (3). h

Although there are many other possibilities following a similar idea and applying symmetry, in order to define new ad-joint triples from another one, we will only use the properties presented above.

Now, we continue reminding the multi-adjoint concept lattice framework from [22]. In the following definition we pres-ent the basic structure which allows the existence of several adjoint triples with respect to L1, L2, P, where (L1,�1) and (L2,�2)are complete lattices. L1 and L2 are required to be lattices since the supremum on L1 and L2 are considered in the definition ofthe generalization of the sufficiency operator, that will be introduced later, and a top element is needed. Moreover, L1 and L2

must be complete lattices since the set of attributes A and objects B, that will be considered, could be infinite. Moreover, con-sidering different adjoint triples will add more flexibility to the language. For example, it was showed that they contribute todescribe preference among objects or attributes.

Definition 6. A multi-adjoint frame L is a tuple

ðL1; L2; P;�1;�2;6;&1;.1;-1; . . . ;&n;.n;-nÞ

where (L1,�1) and (L2,�2) are complete lattices, (P,6) is a poset and, for all i = 1, . . . ,n, (&i,.i,-i) is an adjoint triple withrespect to L1, L2, P.

Multi-adjoint frames are denoted as (L1,L2,P,&1, . . . ,&n).Given a frame, a multi-adjoint context is a tuple consisting of sets of objects, attributes and a fuzzy relation among them; in

addition, the multi-adjoint approach also includes a function which assigns an adjoint triple to each object (or attribute).

Definition 7. Let (L1,L2,P,&1, . . . ,&n) be a multi-adjoint frame, a context is a tuple (A,B,R,r) such that A and B are non-emptysets (usually interpreted as attributes and objects, respectively), R is a P-fuzzy relation R :A � B ? P and r :B ? {1, . . . ,n} is amapping which associates any element in B with some particular adjoint triple in the frame.1

milar theory could be developed by considering a mapping s : A ? {1, . . . , n} which associates any element in A with some particular adjoint triple in the


The set of mappings g :B ? L2, f :A ? L1 will be noted as usual LB2 and LA

1, respectively. On these sets a pointwise partialorder can be considered from the partial orders in (L1,�1) and (L2,�2), which provides LB

2 and LA1 with the structure of com-

plete lattice, that is, abusing notation, LB2;�2

� �and LA

1;�1

� �are complete lattice where �2 is defined pointwise, given

g1; g2 2 LB2; f 1; f 2 2 LA

1; g1�2 g2 if and only if g1(b) �2 g2(b), for all b 2 B; and f1 �1 f2 if and only if f1(a) �1 f2(a), for all a 2 A.Once we have fixed a multi-adjoint frame and a context for that frame, we can define the following mappings "r : LB

2 ! LA1

and #r : LA1 ! LB

2, which generalize the classical definitions given in (1), (2), and that can be seen as generalisations of thosefuzzy mappings given in [3,17]:

2 Thi

g"rðaÞ ¼ inffRða; bÞ.rðbÞgðbÞj b 2 Bg ð6Þf #

r ðbÞ ¼ inffRða; bÞ-rðbÞf ðaÞj a 2 Ag ð7Þ

It is not difficult to show that these two arrows generate a Galois connection [22].The notion of concept is defined as usual: a multi-adjoint concept is a pair hg, fi satisfying that g 2 LB

2; f 2 LA1 and that g"r ¼ f

and f #r ¼ g; with ("r ; #r) being the Galois connection defined above.

Finally, the definition of concept lattice in this framework is defined [22].

Definition 8. The multi-adjoint concept lattice associated to a multi-adjoint frame (L1,L2,P,&1, . . . ,&n) and a context (A,B,R,r)is the set

M¼ fhg; f ij g 2 LB2; f 2 LA

1 and g"r ¼ f ; f #r ¼ gg

in which the ordering is defined by hg1, f1i � hg2, f2i if and only if g1 �2 g2 (equivalently f2 �1 f1).In [22], the authors proved that the ordering just defined above provides M with the structure of a complete lattice.

Moreover, a representation theorem to multi-adjoint concept lattices was proved, which generalizes the classical one andsome other fuzzy generalizations.

In the following sections we will generalize the definitions of the necessity and possibility operators in a fuzzy environ-ment, in a similar way that the fuzzy definitions of the sufficient operators were given in Eqs. (6) and (7). Moreover, newconcept lattices will be presented.

4. Multi-adjoint property-oriented concept lattice

First of all, the frame in this environment must be defined. Given two complete lattices (L1,�1) and (L2,�2), a poset (P,6)and adjoint triples with respect to2 P, L2, L1, (&i,.i,-i), for all i = 1, . . . ,n, a multi-adjoint property-oriented frame is the tuple

ðL1; L2; P;�1;�2;6;&1;.1;-1; . . . ;&n;.n;-nÞ

Multi-adjoint property-oriented frames are denoted as (L1,L2,P,&1, . . . ,&n). L1 and L2 need to be lattices since the infimum andsupremum on L1 and L2 are considered in the definitions of "p and ;N

, which will be introduced later. Moreover, L1 and L2 mustbe complete lattices since the sets of attributes A and objects B, that will be assumed in the context, could be infinite.

Note that the notation is similar to a multi-adjoint frame, although the adjoint triples are defined on different carriers. Thesame order for L1, L2 and P as in a multi-adjoint frame is considered because the attributes, objects and the relation are asso-ciated to L1, L2 and P, respectively, as in the multi-adjoint concept lattices environment.

The definition of context is analogous to the one given in the previous section. Given a multi-adjoint property-orientedframe, (L1,L2,P,&1, . . . ,&n), a context is a tuple (A,B,R,r) such that A and B are non-empty sets (usually interpreted as attri-butes and objects, respectively), R is a P-fuzzy relation R :A � B ? P and r :B ? {1, . . . ,n} is a mapping which associatesany element in B with some particular adjoint triple in the frame.

From now on, we will fix a multi-adjoint property-oriented frame and context, (L1,L2,P,&1, . . . ,&n), (A,B,R,r). Now, giveng 2 LB

2, and f 2 LA1, we define the following mappings: "p : LB

2 ! LA1;#N

: LA1 ! LB

2:

g"pðaÞ ¼ supfRða; bÞ&bgðbÞj b 2 Bg ð8Þf #

N ðbÞ ¼ infff ðaÞ-bRða; bÞj a 2 Ag ð9Þ

Clearly, these definitions are generalizations of the classical possibility and necessity operators. Moreover, we will prove thatthe mappings "p : LB

2 ! LA1;#N

: LA2 ! LB

1 lead us to build a lattice which generalize the concept lattice introduced in [6] to afuzzy environment. Consequently, the properties given in [13,6,30] can be presented in this general framework.

First of all, as usual, we introduce the notion of concept, in this environment, as a pair of mappings hg, fi, with g 2 LB, f 2 LA,such that g"p ¼ f and f #

N ¼ g, which will be called multi-adjoint property-oriented formal concept. The set of all these conceptswill be denoted as MpN.

As a consequence of the following result, the set of all concepts, MpN, forms a complete lattice with the ordering:hg1, f1i 6 hg2, f2i if and only if g1 �2 g2, or equivalently, if and only if f1 �1 f2.

s is the main different with respect to the multi-adjoint frame, as it will be commented below.


Lemma 2. If we consider the posets ðL1;�op1 Þ; ðL2;�2Þ and (P,6op) instead of (L1,�1), (L2,�2) and (P,6), then we obtain that the

pair ("p, ;N

) is equal to the pair (", ;) given in Eqs. (6) and (7).

Proof. From Lemma 1(2), for each i 2 {1, . . . ,n}, as (&i,.i,-i) is an adjoint triple with respect to (P,6), (L2,�2) and (L1,�1), thetriple (.i,&i,-i,op) is an adjoint triple with respect to ðL1;�op

1 Þ; ðL2;�2Þ and (P,6op), where -i,op is the opposite operator of-i. Hence, ðL1; L2; P;�op

1 ;�2;6op;.1; . . . ;.nÞ is a multi-adjoint frame and the mappings ("p, ;

N

) are defined, for all a 2 A andb 2 B, as:

g"pðaÞ ¼ sup1fRða; bÞ&bgðbÞj b 2 Bg

¼ inf1;opfRða; bÞ&bgðbÞj b 2 Bg

¼ð�Þ g"ðaÞ

f #N ðbÞ ¼ inf2ff ðaÞ-bRða; bÞj a 2 Ag

¼ inf2fRða; bÞ-b;opf ðaÞj a 2 Ag

¼ð��Þ f #ðbÞ

where sup1 is the supremum on (L1,�1), inf1,op is the infimum on ðL1;�op1 Þ; inf2 is the infimum on (L2,�2) and the equality (⁄)

follows from Lemma 1(2). Since (.b,&b,-b,op) is an adjoint triple with respect to L@1; L2 and P@, &b plays the role of the impli-cation used in the definition of g", given in Eq. (6). Equality (⁄⁄) is given because-b,op is the implication used in the definitionof f #, given in Eq. (7).

Thus, the pair ("p, ;N

), defined in the multi-adjoint property-oriented frame (L1,L2,P,�1,�2,6,&1, . . . ,&n), is equal to the pair(", ;), defined in the multi-adjoint frame ðL1; L2; P;�op

1 ;�2;6op;.1; . . . ;.nÞ. h

Consequently, we can transfer the properties given in multi-adjoint concept lattices to multi-adjoint property-orientedconcept lattices, as, for example, the following result.

Theorem 1. The pair (MpN,6) is a complete lattice where

inffhgi; fiiji 2 Ig ¼ inf2fgij i 2 Ig; ðinf1ffij i 2 IgÞ#N"p

D E

supfhgi; fiiji 2 Ig ¼ ðsup2fgij i 2 IgÞ"p#N

; sup1ffij i 2 IgD E

Proof. From Lemma 2 we have that ("p, ;N

) is a Galois connection on ðL1;�op1 Þ and (L2,�2), therefore, applying the result given

in [22], we have that the set MpN forms a complete lattice with the order 6op defined as: hg1, f1i 6op hg2, f2i if and only ifg1 �2 g2, or equivalently, if and only if f2�op

1 f1, where:

inffhgi; fiij i 2 Ig ¼ inf2fgij i 2 Ig; ðsup1;opffij i 2 IgÞ#N"p

D E

supfhgi; fiij i 2 Ig ¼ ðsup2fgij i 2 IgÞ"p#N

; inf1;opffij i 2 IgD E

such that sup1,op and inf1,op are the supremum and infimum on L1;�op1

� �, respectively. Thus, changing (L1,�1) by L1;�op

1

� �we

obtain the result. h

The complete lattice (MpN,6) is called multi-adjoint property-oriented formal concept lattice.As a consequence of Lemma 2, we also have that the composition mapping "p;

N

: (L2�2) ? (L2� 2) is a closure operator and;N"p : (L1�1) ? (L1�1) is an interior operator, which is very important in order to obtain the elements of (MpN,6).

4.1. The representation theorem

In [22], the authors presented an extension of the representation (or fundamental) theorem on the classical concept lat-tice [12] for the multi-adjoint framework. In this section, a generalization of the multi-adjoint property-oriented frameworkwill be introduced transforming the results given in this paper.

Some definitions and technical results must be introduced in order to obtain a self-contained paper. First of all, the def-inition of infimum-dense (resp. supremum-dense) subset K # L will be presented; later a definition and a proposition will beextracted from [22].

Definition 9. Given a complete lattice L, a subset K # L is infimum-dense (resp. supremum-dense) if and only if for all x 2 Lthere exists K0 # K such that x = inf(K0) (resp. x = sup(K0)).


The following definition is fundamental to the representation theorem of multi-adjoint concept lattices.

Definition 10 [22]. Let (L1,L2,P,&1, . . . ,&n) be a multi-adjoint frame and (A,B,R,r) a context. A multi-adjoint concept latticeðM;�Þ is represented by a complete lattice (V,v) if there exists a pair of mappings a : A � L1 ? V and b :B � L2 ? V such that:

(1a) a[A � L1] is infimum-dense;(1b) b[B � L2] is supremum-dense; and

(2) For each a 2 A, b 2 B, x 2 L1 and y 2 L2:

bðb; yÞ v aða; xÞ if and only if x&by 6 Rða; bÞ

From the definition above, some properties can be obtained which were used to prove the representation theorem to themulti-adjoint framework.

Proposition 1 [22]. Given a multi-adjoint frame (L1,L2,P,&1, . . . ,&n), a context (A,B,R,r), a complete lattice (V,v) whichrepresents a multi-adjoint concept lattice ðM;�Þ, and mappings f 2 LA

1 and g 2 LB2, we have:

1. b is order-preserving in the second argument.2. a is order-reversing in the second argument.3. g"(a) = sup{x 2 L1j vg v a(a,x)}, where vg = sup{b(b,g(b))j b 2 B}.4. f;(b) = sup{y 2 L2j b(b,y) v vf}, where vf = inf{a(a, f(a))j a 2 A}.5. If gv(b) = sup{y 2 L2j b(b,y) v v}, then sup{b(b,gv(b))j b 2 B} = v.6. If fv(a) = sup{x 2 L1j v v a(a,x)}, then sup{a(a, fv(a))j a 2 A} = v.

The representation theorem was presented.

Theorem 2 [22]. Given a multi-adjoint frame (L1,L2,P,&1, . . . ,&n) and a context (A,B,R,r); a complete lattice (V,v) represents amulti-adjoint concept lattice ðM;�Þ if and only if (V,v) is isomorphic to ðM;�Þ.

Now, in order to obtain similar results to the multi-adjoint property-oriented framework, we need to introduce the def-initions above in the corresponding property-oriented framework.

Definition 11. Let (L1,L2,P,&1, . . . ,&n) be a multi-adjoint property-oriented frame and a context is (A,B,R,r). A multi-adjointproperty-oriented concept lattice ðMpN;�Þ is represented by a complete lattice (V,v) if there exists a pair of mappingsa :A � L1 ? V and b :B � L2 ? V such that:



bðb; yÞ v aða; xÞ if and only if Rða; bÞ 6 x.by

Note that the definition above has been given by rewriting Definition 10, considering the lattices L@1; L2 and P@. In thiscase, (.i,&i,-i,op) are adjoint triples on L@1; L2 and P@, therefore,.i plays the role of the conjunctor, and the order on P mustbe changed. Thus, the formula R(a,b) 6 x.by is considered instead of the original one, x&y 6 R(a,b).

The following proposition is obtained by Proposition 1, considering the lattices L@1; L2 and P@.

Proposition 2. Let (L1,L2,P,&1, . . . ,&n) be a multi-adjoint property-oriented frame and (A,B,R,r) a context. A complete lattice(V,v) which represents a multi-adjoint concept lattice ðMpN;�Þ, and mappings f 2 LA


1. b is order-preserving in the second argument.2. a is order-preserving in the second argument.3. g"pðaÞ ¼ inf1fx 2 L1j vg v aða; xÞg, when vg = sup{b(b,g(b))j b 2 B}.4. f #N ðbÞ ¼ sup2fy 2 L2j bðb; yÞ v v f g, if vf = inf{a(a, f(a))j a 2 A}.5. If gv(b) = sup 2{y 2 L2j b(b,y) v v}, then sup{b(b,gv(b))j b 2 B} = v.6. If fv(a) = inf 1{x 2 L1j v v a(a,x)}, then sup{a(a, fv(a))j a 2 A} = v.

Proof. As we explained above, ðL@1; L2; P@ ;.1; . . . ;.nÞ is a multi-adjoint frame. Therefore, Proposition 1 rewritten in this

frame leads us to the proof of this proposition. h

Thus, the fundamental theorem for the multi-adjoint property-oriented concept lattices is similar to Theorem 2.


Theorem 3. Let (L1,L2,P,&1, . . . ,&n) be a multi-adjoint property-oriented frame and (A,B,R,r) a context. A complete lattice (V,v)represents a multi-adjoint property-oriented concept lattice ðMpN;�Þ if and only if (V,v) is isomorphic to ðMpN;�Þ.

5. Multi-adjoint object-oriented concept lattice

Following the ideas of the previous section, the multi-adjoint object-oriented concept lattice can be introduced, whichgeneralizes, to a fuzzy environment, the concept lattice introduced in [6]. Given two complete lattices (L1,�1) and (L2,�2),a poset (P,6) and adjoint triples with respect to L1, P, L2, (&i,.i,-i), for all i = 1, . . . ,n, a multi-adjoint object-oriented frameis the tuple

ðL1; L2; P;�1;�2;6;&1;.1;-1; . . . ;&n;.n;-nÞ

Multi-adjoint object-oriented frames are denoted as (L1,L2,P,&1, . . . ,&n). Note that the notation is similar to a multi-adjointframe, although the adjoint triples are defined on different carriers, now they are defined on L1,P and L2, which will be the keyto apply Lemma 1 to relate this frame to the multi-adjoint frame.

As in the previous section, the same order for L1, L2 and P as in a multi-adjoint frame is considered because the attributes,objects and the relation are associated to L1, L2 and P, respectively, as in the multi-adjoint concept lattice environment.

Given a multi-adjoint object-oriented frame, (L1,L2,P,&1, . . . ,&n), a context is a tuple (A,B,R,r) such that A and B are non-empty sets, R is a P-fuzzy relation R :A � B ? P and r :B ? {1, . . . ,n} is a mapping which associates any element in B withsome particular adjoint triple in the frame. The mappings "N : LB

2 ! LA1;#p : LA

1 ! LB2 are defined as:

g"N ðaÞ ¼ inffgðbÞ.bRða; bÞj b 2 Bg ð10Þf #

p ðbÞ ¼ supff ðaÞ&bRða; bÞj a 2 Ag ð11Þ

Considering these mappings we obtain a lattice which generalizes the object-oriented concept lattice introduced in [6] to afuzzy environment. Moreover, as in the case above, the properties given in [13,6,30] can be presented in this generalframework.

Hence, a multi-adjoint object-oriented formal concept is a pair of mappings hg, fi, with g 2 LB, f 2 LA, such that g"N ¼ f andf #

p ¼ g.Analogously as above, but considering ðL1;�1Þ; L2;�op

2

� �and (P,6op) instead of (L1,�1), (L2,�2) and (P,6), we can prove,

using Lemma 1(4), that the pair ("N, ;p), defined in the multi-adjoint object-oriented frame (L1,L2,P,�1,�2,6,&1, . . . ,&n), is

equal to the pair (",;), defined in the multi-adjoint frame ðL1; L2; P;�1;�op2 ;6

op;-1;op; . . . ;-n;opÞ.

Lemma 3. If we consider the posets ðL1;�1Þ; L2;�op2

� �and (P,6op) instead of (L1,�1), (L2,�2) and (P,6), then we obtain that the

pair ("N, ;p) is equal to the pair (", ;) given in Eqs. (6) and (7).

Proof. From Lemma 1, for each i 2 {1, . . . ,n}, as (&i,.i,-i) is an adjoint triple with respect to (L1,�1), (P,6) and (L2,�2), thetriple -i;op;.i;op;&op

i

� �is an adjoint triple with respect to ðL1;�1Þ; ðL2;�op

2 Þ and (P,6op), where-i;op; .i;op; &opi are the oppo-

site operators of-i, .i, &i, respectively. Hence, ðL1; L2; P;�1;�op2 ;6

op;-1;op; . . . ;-n;opÞ is a multi-adjoint frame and the map-pings ("N, ;

p) are defined, for all a 2 A and b 2 B, as:

g"N ðaÞ ¼ inf1fgðbÞ.bRða; bÞj b 2 Bg ¼ inf1fRða; bÞ.b;opgðbÞj b 2 Bg ¼ð�Þ g"ðaÞ

f #p ðbÞ ¼ sup2ff ðaÞ&bRða; bÞj a 2 Ag ¼ inf2;opfRða; bÞ&op

b f ðaÞj a 2 Ag ¼ð��Þ f #ðbÞ

where inf1 is the infimum on (L1,�1), sup2 is the supremum on (L2,�2) and inf2,op is the infimum on ðL2;�op2 Þ and the equality

(⁄) follows from Lemma 1(2). Since -b;op;.b;op;&opb

� �is an adjoint triple with respect to L1; L@2 and P@,.b,op is the implication

used in the definition of g", given in Eq. (6). Equality (⁄⁄) is given because &opb plays the role of the implication used in the

definition of f #, given in Eq. (7).Thus, the pair (;

p, "N), defined in the multi-adjoint property-oriented frame (L1,L2,P,�1,�2,6,&1, . . . ,&n), is equal to the pair

(", ;), defined in the multi-adjoint frame ðL1; L2; P;�1;�op2 ;6

op;-1;op; . . . ;-n;opÞ. h

Consequently, as in the property-oriented case, we can transfer the properties given in multi-adjoint concept lattices tomulti-adjoint object-oriented concept lattices, as, for example, that the set of all multi-adjoint object-oriented formal con-cepts, denoted as MNp, with the order: hg1, f1i 6 hg2, f2i if and only if g1 �2 g2, or equivalently, if and only if f1 �1 f2, form acomplete lattice, which is called multi-adjoint object-oriented formal concept lattice. That is:

Theorem 4. The pair (MNp,6) is a complete lattice where

inffhgi; fiij i 2 Ig ¼ sup2fgij i 2 Ig; ðsup1ffij i 2 IgÞ#p"N

D E

supfhgi; fiij i 2 Ig ¼ ðinf2fgij i 2 IgÞ"N#p ; inf1ffij i 2 IgD E


Moreover, the composition mapping "N;p: (L2�2) ? (L2�2) is an interior operator and ;p"N : (L1�1) ? (L1�1) is an closureoperator, which is very important in order to obtain the elements of (MNp,6).

5.1. The representation theorem

The extension of the representation (or fundamental) theorem on the classical concept lattice [12] for the multi-adjointobject-oriented framework is similar to the one given in the previous section to multi-adjoint property-oriented conceptlattices.

Now, the meaning of that lattice which represents a multi-adjoint object-oriented concept lattice will be presented.

Definition 12. Let (L1,L2,P,&1, . . . ,&n) be a multi-adjoint object-oriented frame and a context is (A,B,R,r). A multi-adjointobject-oriented concept lattice ðMpN;�Þ is represented by a complete lattice (V,v) if there exists a pair of mappingsa :A � L1 ? V and b :B � L2 ? V such that:



bðb; yÞ v aða; xÞ if and only if Rða; bÞ 6 y-bx

Note that the definition above is similar to Definition 10. Considering the lattices L1; L@2 and P@, we have that (.i,&i,-i,op)are adjoint triples on L1; L@2 and P@. Therefore, as the implications.i play the role of the conjunctors, and the order on P is thedual, the original formula, x&by 6 R(a,b), is, in this case, R(a,b) 6 x.by.

The transformation of Proposition 1 to this environment is the following:

Proposition 3. Let (L1,L2,P,&1, . . . ,&n) be a multi-adjoint property-oriented frame and (A,B,R,r) a context. A complete lattice(V,v) which represents a multi-adjoint concept lattice ðMpN;�Þ, and mappings f 2 LA


1. b is order-reversing in the second argument.2. a is order-reversing in the second argument.3. g"p ðaÞ ¼ sup1fx 2 L1j vg v aða; xÞg, if vg = sup{b(b,g(b))j b 2 B}.4. f #N ðbÞ ¼ inf2fy 2 L2j bðb; yÞ v v f g, when vf = inf{a(a, f(a))j a 2 A}.5. If gv(b) = inf 2{y 2 L2j b(b,y) v v}, then sup{b(b,gv(b))j b 2 B} = v.6. If fv(a) = sup1 {x 2 L1j v v a(a,x)}, then sup{a(a, fv(a))j a 2 A} = v.

Finally, the fundamental theorem for multi-adjoint object-oriented concept lattices is the following.

Theorem 5. Given a context (A,B,R,r) on a multi-adjoint object-oriented frame (L1,L2,P,&1, . . . ,&n). A complete lattice (V,v)represents a multi-adjoint object-oriented concept lattice ðMNp;�Þ if and only if (V,v) is isomorphic to ðMNp;�Þ.

6. A worked example

Let us consider a set of countries (objects) and a set of polluting gases (attributes) and we need to know what countrycontaminates the least, some values will be fixed as a threshold which must not be surpassed. The data will be similar tothe given in the example presented in [22] in order to compare the results.

We will consider a multi-adjoint frame with three different lattices: one for handling the information of the amount of gasdischarged, which is rounded to the second decimal digit; a second one to handle information about the attributes, in whichwe estimate steps of 0.05 in order to distinguish to appreciate a qualitative difference; and a third one, used to set the dif-ferent levels of preference of the countries, which is considered to be of 0.125 (hence the unit interval is divided into eightequal pieces).

The multi-adjoint property-oriented frame considered is

½0;1�20; ½0;1�8; ½0;1�100;6;6;6;&�P ;&

�L

� �

where [0,1]m denotes a regular partition of [0,1] into m pieces, for each m 2 N, and &�P; &�L are the discretizations of the prod-uct and Łukasiewicz conjunctors respectively, defined, for all a 2 [0,1]100, b 2 [0,1]8, as:

&�Pða; bÞ ¼d20 � a � be

20; &�Lða; bÞ ¼

d20 �maxfaþ b� 1;0ge20

where d_e is the ceiling function. The corresponding residuated implications .�P;.�L : ½0;1�20 � ½0;1�8 ! ½0;1�100 and-�P;-

�L : ½0;1�20 � ½0;1�100 ! ½0;1�8 are defined as:

Table 1Fuzzy r

R

g1

g2

g3

g4


b.�Pa ¼ b100 �minf1; b=agc100

; b-�Pc ¼ b8 �minf1; b=cgc8

b.�La ¼ b100 �minf1;1þ b� agc100

; b-�Lc ¼ b8 �minf1;1þ b� cgc8

where b_c is the floor function.The multi-adjoint context is given by the set of gases A = {g1,g2,g3,g4}, the set of objects B = {C1,C2, . . . ,C7} and the fuzzy

relation R :A � B ? [0,1]100 defined in Table 1.The problem of choosing the country which contaminates the least depends on the definition of ‘‘contamination’’. For

example, some values will be fixed as a threshold which must not be surpassed, hence, if some country passes some of thisthreshold then it will be penalized in the calculation.

These values can be: 0.75 for gas g1, 0.3 for g2, 0.25 for g3, and 0.5 for g4. Thus, the mapping that represents this notion ofthe best country is the fuzzy subset f :A ? [0,1], defined as:

f ðg1Þ ¼ 0:75; f ðg2Þ ¼ 0:3; f ðg3Þ ¼ 0:25; f ðg4Þ ¼ 0:5

Moreover, as any preference has been assumed, the context (A,B,R,r), where r(b) = &P for every b 2 B, is considered.Therefore, the best country will be given by the multi-adjoint property-oriented concept obtained from the fuzzy set f.

Specifically, the component f #N

will provide required information about each country.For example, the computation of f;(C1) is:

f #N ðC1Þ ¼ infff ðaÞ-�PRða;C1Þj a 2 Ag ¼ inff0:75-�P0:34;0:3-�P0:13;0:25-�P0:31;0:5-�P0:75g

¼ b8 �minf1;0:5=0:75gc8

¼ 0:625

The computation is similar for the rest of the countries, obtaining the following results:

f #N ðC1Þ ¼ 0:625; f #

N ðC2Þ ¼ 0:25; f #N ðC3Þ ¼ 0:25; f #

N ðC4Þ ¼ 0:375

f #N ðC5Þ ¼ 0:25; f #

N ðC6Þ ¼ 0:5; f #N ðC7Þ ¼ 0:25

Therefore, the country with less contamination is C1. Note that C1 has not associated the value 1 because the value to g3 andto g4 surpasses the threshold given by f. Specifically, the associate value C1, which is 0.625, is obtained by g4.

As in the multi-adjoint concept lattice framework presented in [22], one important feature is that preference on subsets ofobjects or attributes may be considered. This is possible since different adjoint triples can be associated with each object(resp. attribute). For instance, if some countries have less possibilities of decreasing contamination due to their third worldcondition, we can modify the underlying context in order to assign preference to these countries, saying we would preferwhether one of these countries won our contest, although this preferability is not an imposition.

For example, C4, C6 and C7 are marginal countries and they do not have the same possibilities as the rest of countries ofdecreasing pollution, hence we will prefer one of these countries to win.

In order to assume this preference, we consider the context (A,B,R,r0), where r0(b) = &P for every b 2 B1 and r0(b) = &L forevery b 2 B2, where B1 = {C1,C2,C3,C5} and B2 = {C4,C6,C7}.

This particular selection of r0 allows the use of the Łukasiewicz implication in order to compute the values for marginalcountries, hence the definition of f #

N is modified considering different cases:

f #N ðb1Þ ¼ inffRða; b1Þ-�Pf ðaÞja 2 Ag for b1 2 B1

f #N ðb2Þ ¼ inffRða; b2Þ-�Lf ðaÞja 2 Ag for b2 2 B2

In this case, we obtain that

f #N ðC1Þ ¼ 0:625; f #

N ðC2Þ ¼ 0:25; f #N ðC3Þ ¼ 0:25; f #

N ðC4Þ ¼ 0:5

f #N ðC5Þ ¼ 0:25; f #

N ðC6Þ ¼ 0:75; f #N ðC7Þ ¼ 0:25

Therefore, the best country is C6 which is an element of the preference set. Despite this, the winner is not always in the pref-erence subset. For example, consider the following modification f1 of the notion of the best country:

elation between the objects and the attributes.

C1 C2 C3 C4 C5 C6 C7

0.34 0.21 0.52 0.85 0.43 0.21 0.090.13 0.09 0.36 0.17 0.1 0.04 0.060.31 0.71 0.92 0.65 0.89 0.47 0.930.75 0.5 1 1 0.5 0.25 0.25


f1ðImpact FactorÞ ¼ 0:75; f 1ðImmediacy IndexÞ ¼ 0:3;f1ðCited Half � LifeÞ ¼ 0:2; f 1ðBest PositionÞ ¼ 0:5

In this case, the results associated to this f1 are

f #N ðC1Þ ¼ 0:625; f #

N ðC2Þ ¼ 0:25; f #N ðC3Þ ¼ 0:125; f #

N ðC4Þ ¼ 0:5

f #N ðC5Þ ¼ 0:125; f #

N ðC6Þ ¼ 0:625; f #N ðC7Þ ¼ 0:25

which states that the countries that best suit our needs are C1 and C6. In this case, a preferable country did not win. Indeed,examples can be given in order to show that the preferred countries are not among the better ones. We only need to consideranother subset of the countries that are preferable.

Therefore, the flexibility of the framework presented provides a wide number of possible applications. The example aboveshows that the multi-adjoint property-oriented concept lattices may be used in problems where a threshold cannot be sur-passed, such as the pollution problem of the example; economic problems about budget, construction of buildings; diagnosisproblems where it is necessary to analyze a patient whose levels of cholesterol, sugar, bilirubin, . . . , should not surpass thethresholds; etc.

Comparison with multi-adjoint concept lattices. Now, we will compare this example of multi-adjoint property-orientedconcept lattices with the framework presented in [22]. Specifically, given a fuzzy subset of attributes f, we will check thatthe concepts obtained in both frameworks, ðf #N"p ; f #

N Þ, (f #"; f #), are not dual. In order to do that the context of the previousexample will be considered.

The multi-adjoint frame and the multi-adjoint property-oriented frame will be equals, for that, P = L1 must be assumed, inorder to consider the same definitions of adjoint triples in both frames, that is,

½0;1�100; ½0;1�8; ½0;1�100;6;6;6;&�P;&

�L

� �

where &�P is the discretisation of the product conjunctor, defined for all a 2 [0,1]100, b 2 [0,1]8, as:

&�Pða; bÞ ¼d100 � a � be

100

where d_e is the ceiling function. The corresponding residuated implications .�P : ½0;1�100 � ½0;1�8 ! ½0;1�100 and-�P : ½0;1�100 � ½0;1�100 ! ½0;1�8 are defined as:

b.�Pa ¼ b100 �minf1; b=agc100

; b-�Pc ¼ b8 �minf1; b=cgc8

where b_c is the floor function.If we consider the fuzzy subset f :A ? [0,1] below:

f ðg1Þ ¼ 0:75; f ðg2Þ ¼ 0:3; f ðg3Þ ¼ 0:55; f ðg4Þ ¼ 0:5

The computation of the extension f; is:

f #ðC1Þ ¼ 0:625; f #ðC2Þ ¼ 0:75; f #ðC3Þ ¼ 0:5; f #ðC4Þ ¼ 0:5

f #ðC5Þ ¼ 0:5; f #ðC6Þ ¼ 1; f #ðC7Þ ¼ 0:5

And the computation of the extension f #N is:

f #N ðC1Þ ¼ 0:375; f #

N ðC2Þ ¼ 0:25; f #N ðC3Þ ¼ 0:625; f #

N ðC4Þ ¼ 0:5

f #N ðC5Þ ¼ 0:25; f #

N ðC6Þ ¼ 0:125; f #N ðC7Þ ¼ 0

Hence, the values given by f #andf #N

are not dual, indeed the order among the countries given by the values of these mappingsare not dual. For example, the best country by f # is C2 but the worst country by f #

Nis C7.

7. Conclusions and future work

The philosophy of the multi-adjoint paradigm [24,16] was used to generalise the property-oriented and object-orientedconcept lattices [6], which themselves embedded the rough set theory [11]. Moreover, among others, these generalisationsembed the fuzzy antitone (isotone) Galois connection provided by Georgescu and Popescu in [14] and the concept lattices offuzzy contexts based on rough set theory provided by Lai and Zhang in [18].

Therefore, we can consider a general framework in a non-commutative environment, which lead us naturally to considerseveral adjoint triples defined on non-linear sets; this allows us to easily establish different degrees of preference among thesets of objects or attributes.

The main feature of these generalisations is that they directly relate to the multi-adjoint concept lattice environment, inwhich negations are not needed. Thus, the results provided by these frameworks can be transferred from one to the other.


For example, we proved that the multi-adjoint property (object)-oriented concept lattice is indeed a lattice and we alsoproved the representation theorems by using the results given for the multi-adjoint concept lattices in [22].

Moreover, a detailed example was introduced in which several important features of the multi-adjoint property-orientedconcept lattice, such as the possibility of considering preference among objects or attributes, can be verified.

New results in these new frameworks will be presented as future research and applications to the classical rough set the-ory will be studied. We also wish to study the generalisation of the dual sufficiency operators.

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Multi-adjoint property-oriented and object-oriented concept lattices